Does the sentence imply that it's his own? Wouldn't it then be его папу? In English, without any further qualifier, I'd think it would mean the speaker's father ... although then it also should be capitalized. Спасибо!

Yes, it's the affectionate/short form of Dmitri.

I too am unclear on whether this is better translated to:

Dima doesn't know a father. (meaning he's never known what it's like to have a dad)

or

Dima doesn't know Dad. (meaning you're talking about your own father, no relation to Dima)

Can any native speaker help?

It looks like it's about Dima's father, as if they were talking about not his father, they would call this man just a man or they would definitely say whose father Dima does not know, and if there is no specification, obviously it is about his father. (Sorry if I have mistakes.)

I don't think so, it seems to be like English in this regard. (Though other languages e.g. German have wissen / kennen in a similar way to the French example you cite.) (Disclaimer: I am not a native Russian!)

FEI (For Everyone's Information): In short form, savoir is to know things, connaitre is to know people. See:
https://www.thoughtco.com/savoir-vs-connaitre-1368940

The words are often cut off in the slow versions. It is extremely time-consuming and difficult to edit sound down to one word, especially when a word is being extracted from a string of words (a spoken sentence). If it's too long, you often get pieces of the previous or subsequent word mixed. Getting a perfect sound for the slow track would be extremely expensive and time-consuming. I rely on the slow version to get the basic word down, but listen to the fast track for endings. Sometimes you just have to make an educated guess, usually based on the module you're studying.

This is true of every language on Duo - and in other language programs, too. Often, you get what you pay for.